You could paint it, and this looks great when fresh, but it will need maintenance. I would start on the garden and painting the brick wall a nicer colour. Add some big planter boxers onto the concrete and more layering in the beds and let it mature. The little kids will love riding on the concrete and doing chalk drawings etc. you could paint some games like hopscotch and 4 square. Don’t rush. Garden first

Kate, what colour would you suggest? Not sure I understand when you say 'garden first'

You can resurface concrete so it looks like exposed aggregate. Don’t have any personal experience with the process, but it might be worth considering. Search for concrete resurfacing in your area. Here is one example:

https://stonecrete.com.au

As Kate said painting is an option, I know you said you didn’t like decking but it could look fabulous in the area you have, tiles or boards would be easy to install. But painting is your cheapest option but needs ongoing maintenance to keep it looking good.

OMG this would look fabulous! If only I had the ca$h. Thanks Julie

Maybe I could do just the 'square' area between the alfresco and that wall? and paing the rest?

Maybe you could paint the concrete, and use a tile stencil with a contrasting colour to paint a "rug" in that square area? If you google stencil concrete rug, there's loads of lovely examples, and tutorials. It's time consuming, but cheap.
From my experience with tile paint: don't skimp on preparing the concrete, otherwise it peels. Also dark colours get very hot, which is a problem with kids. I'd maybe match your grout colour.